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We define a notion of information that an individual sample provides to thetraining of a neural network, and we specialize it to measure both how much asample informs the final weights and how much it informs the function computedby the weights. Though related, we show that these quantities have aqualitatively different behavior. We give efficient approximations of thesequantities using a linearized network and demonstrate empirically that theapproximation is accurate for real-world architectures, such as pre-trainedResNets. We apply these measures to several problems, such as datasetsummarization, analysis of under-sampled classes, comparison of informativenessof different data sources, and detection of adversarial and corrupted examples.Our work generalizes existing frameworks but enjoys better computationalproperties for heavily over-parametrized models, which makes it possible toapply it to real-world networks.

Estimating informativeness of samples with Smooth Unique Information